College Math Placement Test College Math Placement Test Number Theory 2 — Questions and Answers
Question 1: What is the prime factorization of 180?
- 2² × 3² × 5 (Correct answer)
- 2 × 3 × 5²
- 2³ × 3 × 5
- 2² × 3 × 5²
Correct answer: 2² × 3² × 5
180 = 4×45 = 4×9×5 = 2²×3²×5.
Question 2: Which pair of numbers is relatively prime (coprime)?
- 12 and 16
- 15 and 25
- 14 and 27 (Correct answer)
- 18 and 24
Correct answer: 14 and 27
14 = 2×7 and 27 = 3³ share no common factors, so GCD(14, 27) = 1.
Question 3: A number is divisible by 6 if and only if it is:
- Divisible by 2 only
- Divisible by 3 only
- Divisible by both 2 and 3 (Correct answer)
- Divisible by 12
Correct answer: Divisible by both 2 and 3
Since 6 = 2×3 and GCD(2,3) = 1, divisibility by 6 requires divisibility by both 2 and 3.
Question 4: What is the LCM of 6, 10, and 15?
- 30 (Correct answer)
- 60
- 90
- 150
Correct answer: 30
6=2×3, 10=2×5, 15=3×5, so LCM = 2×3×5 = 30.
Question 5: How many positive divisors does 100 have?
- 7
- 8
- 9 (Correct answer)
- 10
Correct answer: 9
100 = 2²×5², so the number of divisors = (2+1)(2+1) = 9.
Question 6: What is the remainder when 2²⁰ is divided by 7?
- 1 (Correct answer)
- 2
- 4
- 6
Correct answer: 1
Powers of 2 mod 7 cycle as 2,4,1 with period 3; since 20=3×6+2, 2²⁰ ≡ 4 (mod 7).
College Math Placement Test College Math Placement Test Number Theory 2
What is the prime factorization of 180?